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Theorem ontric3g 41801
Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥𝑦, 𝑦 = 𝑥, or 𝑦𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
Assertion
Ref Expression
ontric3g 𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g
StepHypRef Expression
1 orcom 869 . . . . . . 7 ((𝑦 = 𝑥𝑦𝑥) ↔ (𝑦𝑥𝑦 = 𝑥))
21a1i 11 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥𝑦𝑥) ↔ (𝑦𝑥𝑦 = 𝑥)))
3 onsseleq 6359 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
4 ontri1 6352 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
52, 3, 43bitr2d 307 . . . . 5 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥𝑦𝑥) ↔ ¬ 𝑥𝑦))
65con2bid 355 . . . 4 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)))
76ancoms 460 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)))
84ancoms 460 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
9 ontri1 6352 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
108, 9anbi12d 632 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦𝑥𝑥𝑦) ↔ (¬ 𝑥𝑦 ∧ ¬ 𝑦𝑥)))
11 eqss 3960 . . . 4 (𝑦 = 𝑥 ↔ (𝑦𝑥𝑥𝑦))
12 ioran 983 . . . 4 (¬ (𝑥𝑦𝑦𝑥) ↔ (¬ 𝑥𝑦 ∧ ¬ 𝑦𝑥))
1310, 11, 123bitr4g 314 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)))
14 equcom 2022 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
1514orbi2i 912 . . . . . 6 ((𝑥𝑦𝑦 = 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦))
1615a1i 11 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑦 = 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦)))
17 onsseleq 6359 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ (𝑥𝑦𝑥 = 𝑦)))
1816, 17, 93bitr2d 307 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑦 = 𝑥) ↔ ¬ 𝑦𝑥))
1918con2bid 355 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
207, 13, 193jca 1129 . 2 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥))))
2120rgen2 3195 1 𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846  w3a 1088  wcel 2107  wral 3065  wss 3911  Oncon0 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322
This theorem is referenced by: (None)
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